A comparative study of fuzzy domination and fuzzy coloring in an optimal approach

Ci a ion: Meenakshi, A.; Kannan, A.; Mahdal, M.; Ka hik, K.; Gu as, R. A Compa a i e S udy o Fuzzy Domina ion and Fuzzy Colo ing in an Op imal App oach. Ma hema ics 2023,11, 4019. h ps://doi.o g/ 10.3390/ma h11184019 Academic Edi o : Kons an in Kozlo Recei ed: 24 July 2023 Re ised: 10 Sep embe 2023 Accep ed: 11 Sep embe 2023 Published: 21 Sep embe 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). ma hema ics A icle A Compa a i e S udy o Fuzzy Domina ion and Fuzzy Colo ing in an Op imal App oach Annamalai Meenakshi 1, Adhimoolam Kannan 1,2 , Mi osla Mahdal 3, K ishnasamy Ka hik 4,* and Radek Gu as 3 1Depa men o Ma hema ics, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and Technology, Chennai 600062, India; [email p o ec ed] (A.M.); [email p o ec ed] (A.K.) 2Depa men o Ma hema ics, Vel Tech Mul i Tech D . Ranga ajan D . Sakun hala Enginee ing College, Chennai 600062, India 3Depa men o Con ol Sys ems and Ins umen a ion, Facul y o Mechanical Enginee ing, VSB-Technical Uni e si y o Os a a, 17. Lis opadu 2172/15, 70800 Os a a, Czech Republic; mi osla [email p o ec ed] (M.M.); [email p o ec ed] (R.G.) 4Depa men o Mechanical Enginee ing, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and Technology, Chennai 600062, India *Co espondence: [email p o ec ed] Abs ac : An op imal ne wo k e e s o a compu e o communica ion ne wo k designed, con ig- u ed, and managed o maximize e iciency, pe o mance, and e ec i eness while minimizing cos and esou ce u iliza ion. In a ne wo k design and managemen con ex , op imal ypically implies achie ing he bes possible ou comes be ween a ious ac o s. This esea ch in es iga ed he use o uzzy g aph edge colo ing o a ious uzzy g aph ope a ions, and i ocused on he e icacy and e iciency o he uzzy ne wo k p oduc using he minimal spanning ee and he ch oma ic index o he uzzy ne wo k p oduc . As a ne wo k made o nodes and e ices, measu emen wi h e ices is a pa ame e o domina ion, and edge measu emen is a pa ame e o edge colo ing, so we used hese wo pa ame e s in he algo i hm. This pape aims o iden i y an op imal ne wo k ha can be es ablished using p oduc ou comes. This s udy shows a way o ind an op imal uzzy ne wo k based on compa a i e op imal pa ame e domina ion and edge colo ing, which can be elabo a ed wi h applica ions. An algo i hm was gene a ed using an op imal app oach, which was subsequen ly implemen ed in he o m o applica ions. Keywo ds: uzzy colo ing; minimum spanning ee; domina ion numbe ; op imal ne wo k MSC: 05C15; 05C76 1. In oduc ion A ma hema ical ool known as g aph heo y plays a i al ole in nume ous b anches o esea ch and echnology. A g aph ypically depic s a eal and ele an p oblem g aphically. A g aph is a collec ion o se s (K,L), whe e Kis a collec ion o non-emp y e ices, and Lis an edge se . Kau man (1973) p esen ed he concep o uzzy g aphs, and u he , Rosen eld (1975) in e p e ed i . Saman a and Pal (2015, 2013) de ined a ious o ms o uzzy g aphs. In he li e a u e, he e a e many ways o colo g aphs. Fuzzy se heo y and uzzy g aph heo y ha e made i possible o model mos eal-wo ld si ua ions mo e p ecisely and adap ably han hei classical coun e pa s [ 1 – 7 ]. Mo e s udy is being conduc ed on uzzy g aphs [ 8 – 10 ]. The usual g aph model o a ne wo k has a collec ion o nodes joined by edges o connec ions. The ne wo k p o ides a lexible amewo k o loca ing and obse ing complex sys ems [ 11 ]. The idea o s udying complex ne wo ks is essen ial and c osses many academic ields. Real-wo ld p oblems con ain a a ie y o da a ha can be ep esen ed using a a ie y o g aph ypes, including uzzy g aphs, in ui ionis ic uzzy g aphs, and Ma hema ics 2023,11, 4019. h ps://doi.o g/10.3390/ma h11184019 h ps://www.mdpi.com/jou nal/ma hema ics Ma hema ics 2023,11, 4019 2 o 16 neu osophic g aphs [ 12 – 17 ]. A i in oduced he concep s o he so o e se and he so o e g aph [ 18 ]. Saman a and o he s p o ided an explana ion o se e al ecen ly c ea ed ideas abou in ui ionis ic uzzy g aphs (IFGs), as well as a ew key concep s ha had al eady been es ablished [ 19 ]. Using he concep s o in ui ionis ic uzzy se s, in ui ionis ic uzzy ela ions, and index ma ices as a ounda ion, a new gene aliza ion o IFGs has been p esen ed [ 20 ]. Many au ho s ha e s udied a ious ypes o domina ions [ 21 – 28 ] and de eloped his ield o s udy. The c isp-g aph colo ing echnique has been used o colo he α -cu s o hese uzzy g aphs. As a esul , many c isp g aphs a e colo ed o di e en alues o α , and o iden ical uzzy g aphs, he ch oma ic index changes depending on he alue o α [ 29 ]. Addi ionally, Be sh ein and Bozhenuk sugges ed using he minimax c i e ion o de ine he ideal cen e alloca ion in uzzy anspo a ion ne wo ks [ 30 ]. One s udy de ined he concep o a uzzy g aph and de e mined he minimum numbe o colo s based on he alue o he sepa a ion deg ee [ 31 – 38 ]. A new colo ing echnique was employed o colo a poli ical map and o add ess a b and-new a ic ligh colo ing issue [ 39 ]. A colo ed e ex on uzzy g aphs was used o colo maps. To o e come adio equency issues, Mahapa a e al. ex ended he colo ing app oach o adio uzzy g aphs [ 40 – 42 ]. The ela ionship (edges) can be mo e meaning ul han he indi idual (nodes) a imes. Fo ins ance, links a he han nodes a e c ucial in uzzy social ne wo ks. An associa ed concep known as edge colo ing is c ucial o issues based on unce ain y. Rega ding he applica ion o g aph colo ing o communica ion sys ems based on u ilizing ing-spli s, in addi ion o compa ing he peak h oughpu o NoCs o ci culan and mesh opologies using deadlock- ee ou ing algo i hms, he esul s o high-le el modeling we e p esen ed in [ 43 ]. The sugges ed me hod used ewe ha dwa e esou ces while s ill achie ing minimal ansmission delay and enhanced he mal e iciency [44]. The mo i a ion o his esea ch wo k is o c ea e a new pla o m o iden i y an op imal ne wo k in o de o de elop an e ec i e op ical ne wo k by u ilizing a ious uzzy g aph ope a ions based on edge colo and domina ion pa ame e s, including he ope a ions o esidue p oduc s, symme ic di e ences, max p oduc s, and lexicog aphic p oduc s. Using compa a i e s udies on domina ion and edge colo ing, ou esea ch and analysis aimed o assess he ne wo k’s s eng h. We ha e p o ided an algo i hm o examine he e ec i eness and e iciency o he cons uc ed ne wo ks, which is he main amewo k o his esea ch inding. Fu he mo e, we ha e de eloped applica ions o he p oduc ope a ion o hese uzzy ne wo ks. 2. P elimina ies De ini ion 1 ([ 12 ]) . A Fuzzy g aph ( H∗FG = (V∗ FG , E∗ FG) ) is a pai o unc ions ( σV∗ FG :V∗ FG →[0, 1] and µV∗ FG :V∗ FG ×V∗ FG →[0, 1] ) whe e µV∗ FG (b1 , b2)≤min{σV∗ FG (b1∗) , σV∗ FG (b2∗)} o b1∗,b2∗∈V∗FG. De ini ion 2 ([ 12 ]) . The unde lying g aph o a uzzy g aph is in he o m ( H∗FG = (V∗ FG , E∗ FG) ), whe e V∗ FG =na1∗∈V∗ FG :σV∗ FG (a1∗)>0o and E∗ FG ={(a1∗ , a2∗)∈V∗ FG ×V∗ FG : µV∗ FG (a1∗,a2∗)>0}. De ini ion 3 ([ 12 ]) . A subse ( T∗ F ) o V∗ FG is said o be a domina ing se o a uzzy g aph i e e y e ex in V∗ FG −T∗ FG is domina ed by a leas one e ex o V∗ FG . The domina ing se ( T∗ FG ) is said o be minimal i no p ope subse o T∗ FG is a domina ing se . De ini ion 4 ([ 12 ]) . An a c ( a1∗ −a2∗ ) is said o be s ong i he alue o deg ee o an edge membe ship o an a c (a1∗ −a2∗) is equal o s eng h o connec edness be ween a1* and a2*. De ini ion 5 ([12]).A e ex a1* domina es a2* i he e is a s ong a c be ween hem. Ma hema ics 2023,11, 4019 3 o 16 De ini ion 6 ([ 34 ]) . A uzzy g aph ηec =(Vec,σec,µec) is a se ha is no emp y, oge he wi h a pai o unc ions σec :Vec →[0, 1] and µec :Vec ×Vec →[0, 1] , such ha x , yeVec , µec(c,d)≤σec(c)Λσec(d) , whe e σec(c) and µec(c,d) ep esen he e ex membe ship alues and he edge membe ship alues, espec i ely. De ini ion 7 [ 35 ]) . A uzzy g aph ηec =(Vec,σec,µec) is comple e i µec(p,q)=min{σec(p) , σec(q)} o all p, q ∈Vec, whe e (p, q) ep esen s he edges be ween he e ices p and q. De ini ion 8 ([ 35 ]) . A uzzy g aph ηec =(Vec,σec,µec) is said o be bipa i e i e ex se Vec is di ided in o wo nonemp y se s Vec1 and Vec2 , such ha µec(Vec1,Vec2)= 0i ec1 , ec2eVec1 o ec1 , ec2eVec2 . Fu he , i µec(Vec1,Vec2)=min{σec( ec1),σec( ec2)} o all ec1eVec1and ec2eVec2 , hen ηec is called a uzzy comple e bipa i e g aph. De ini ion 9 ([29]).Le H = {h1, h2,. . ., hλ}, λ≥1 be he collec ion o neu al hues. Then, uzzy se (H, k), whe e k: H → (0, 1), is known as a collec ion o uzzy colo s, and 0 < k(h i ) ≤ 1; he colo ’s membe ship alue is he quan i y o each elemen o he combina ion o h i wi h he colo whi e. Hence, he colo (h i , k(h i )) is e e ed o as he uzzy colo ha ma ches he undamen al colo h i . Thus, he k(h i ) [ ≤ 1] amoun o h i is mixed wi h 1 − k(h i ) o de e mine how much whi e colo is needed o c ea e he uzzy colo (h i , k(h i )). As s a ed in he de ini ion abo e, he basic colo is he building block om which all o he colo s a e c ea ed. Fo example, g een is a undamen al hue. A “ uzzy g een” colo can be blended o c ea e o he colo s wi h 0.8 uni s o g een and 0.2 uni s o whi e. This “ uzzy g een” is deno ed by a g een alue o 0.8. Simila ly, ano he uzzy ed colo ( ed, 0.6) may be o med by mixing 0.6 uni s o ed wi h 0.4 uni s o whi e, and so on. De ini ion 10 ([ 29 ]) . Le ηec =(Vec,σec,µec) be a connec ed uzzy g aph and cec =cec1,cec2, . . . , ceck be a se o basic colo s. Now, wo edges a e only gi en wo uzzy colo s whose basic colo s di e i hey a e adjacen o one ano he ; o he wise, hey may be gi en uzzy colo s whose basic colo s a e he same. I he colo o any edge is (ceci , ecj(ceci)) , hen Ceci is he basic colo o edge eecj = (p, q) and ecj(ceci) is i s membe ship alue, which is calcula ed as kecj(ceci)=µec(p.q) σec(p)∧σec(q) , whe e σec(p)and σec(q) a e he membe ship alues o e ices p and q, espec i ely. Finally, µec(p,q) is he membe ship alue o he edge eecj , i.e., (p, q) in he uzzy g aph ηec. De ini ion 11 ([ 29 ]) . The uzzy ch oma ic index o a uzzy g aph is he minimal se o undamen al colo s equi ed o colo a uzzy g aph. Suppose he e a e M basic hues a he minimal le el, he s eng hs o edges canno be desc ibed by his ch oma ic index. Fo example, when wo uzzy g aphs ha e iden ical ch oma ic indices, hese g aphs canno be compa ed using his ch oma ic index. Hence, he e is some weigh assigned o he ch oma ic index. The weigh is deno ed by Sec, which is de ined by Sec =∑M i=1nMax eci(ceci)o whe e he basic colo ceci is used o colo edge eecj o some j and he dep h o colo is keci(ceci) . Thus, S is he o al o each basic colo ’s maximum membe ship alues. Now, he ch oma ic index o a uzzy g aph is deno ed by (M, S), whe e M is he minimum numbe o basic colo s o colo a g aph and S is i s weigh . We gene ally ollow he ope a ions on uzzy g aph de ini ions om [13]. 3. Ope a ions on Fuzzy G aphs Using Edge Colo ing Mahaba h a e al. [ 29 ] in oduced De ini ion 11 o de e mine he weigh o colo s in a uzzy g aph. By ocusing on he op imali y o he uzzy ne wo k, we can compa e he o mula gi en in De ini ion 11 wi h he sum o he minimal membe ship alue o each colo used in he uzzy ne wo k. Hence, his esea ch de ines and modi ies he o mula in e ms o he sum o he minimum alue o he edge membe ship alues o each colo acco ding o he uzzy ne wo k, which yields he op imum alue o he c ea ed ne wo k and helps Ma hema ics 2023,11, 4019 4 o 16 us de e mine how e ec i e i is. The ch oma ic numbe o a uzzy g aph ep esen s he minimum numbe o colo s equi ed o colo he g aph. Le us assume ha M min is he minimum numbe o colo s used o colo he g aph. The deg ee o membe ship o such c isp g aphs is no su icien o de e mine he s eng h o edges, and hence, some weigh is associa ed wi h he ch oma ic numbe . These weigh s o he edges can in luence he colo ing p ocess by indica ing he s eng h o associa ion o an edge wi h a pa icula colo . The weigh ed minimum o basic colo s used is deno ed as Wmin and is de ined as Wmin = M ∑ p=1nmin geq(cp)o 3.1. Residue P oduc o Two Fuzzy G aphs Le RF1=(σec1,µec1 ) and RF2=(σec2,µec2) be wo uzzy g aph ne wo ks o c isp g aphs GRF1=(Vec1,Eec1)and GRF2=(Vec2,Eec2) , espec i ely. Then, i s esidue p oduc RF1·RF2=(σ1·σ2,µ1·µ2)is de ined as (i) ∀(a, b) ∈V1×V2, (σ1·σ2)(a,b)=σ1(a)∧σ2(b). (ii) ∀(a, b) ∈E1and c6=w∈V2,(µ1·µ2)((a,c),(b,w)) =µ1(a,b). 3.1.1. Example G 1 =RF 1 and G 2 =RF 2 a e he wo uzzy g aphs o GRF1=(Vec1,Eec1) and GRF2=(Vec2,Eec2) , depic ed in Figu es 1and 2, espec i ely. Then, he esidue p oduc o he uzzy ne wo k is deno ed by RF1·RF2, as shown in Figu e 3. Ma hema ics 2023, 11, x FOR PEER REVIEW 4 o 20 3. Ope a ions on Fuzzy G aphs Using Edge Colo ing Mahaba h a e al. [29] in oduced De ini ion 11 o de e mine he weigh o colo s in a uzzy g aph. By ocusing on he op imali y o he uzzy ne wo k, we can compa e he o mula gi en in De ini ion 11 wi h he sum o he minimal membe ship alue o each colo used in he uzzy ne wo k. Hence, his esea ch de ines and modi ies he o mula in e ms o he sum o he minimum alue o he edge membe ship alues o each colo ac- co ding o he uzzy ne wo k, which yields he op imum alue o he c ea ed ne wo k and helps us de e mine how effec i e i is. The ch oma ic numbe o a uzzy g aph ep esen s he minimum numbe o colo s equi ed o colo he g aph. Le us assume ha M min is he minimum numbe o colo s used o colo he g aph. The deg ee o membe ship o such c isp g aphs is no sufficien o de e mine he s eng h o edges, and hence, some weigh is associa ed wi h he ch oma ic numbe . These weigh s o he edges can in luence he colo ing p ocess by indica ing he s eng h o associa ion o an edge wi h a pa icula colo . The weigh ed minimum o basic colo s used is deno ed as W min and is de ined as {}  = =M pp c q e gW 1 )(min min 3.1. Residue P oduc o Two Fuzzy G aphs Le 𝑅𝐹=(𝜎   ,𝜇  ) and 𝑅𝐹=(𝜎   ,𝜇  ) be wo uzzy g aph ne wo ks o c isp g aphs 𝐺  =𝑉   ,𝐸   and 𝐺  =𝑉   ,𝐸  , espec i ely. Then, i s esidue p oduc 𝑅𝐹∙𝑅𝐹 =(𝜎 ∙𝜎 ,𝜇∙𝜇) is de ined as (i) ∀ (a, b) ∈ V 1 × V 2 , (𝜎∙ 𝜎)(𝑎,𝑏)= 𝜎(𝑎) ∧ 𝜎(𝑏). (ii) ∀ (a, b) ∈ E 1 and c  w ∈ V 2 , (𝜇∙ 𝜇)((𝑎,𝑐),(𝑏,𝑤))= 𝜇 (𝑎,𝑏). 3.1.1. Example G 1 = RF 1 and G 2 = RF 2 a e he wo uzzy g aphs o 𝐺  =𝑉   ,𝐸   and 𝐺  = 𝑉  ,𝐸  , depic ed in Figu e 1 and Figu e 2, espec i ely. Then, he esidue p oduc o he uzzy ne wo k is deno ed by RF 1 ·RF 2 , as shown in Figu e 3. Figu e 1. Fuzzy G aph G 1 . Figu e 2. Fuzzy G aph G 2 . Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed RF 1 ·RF 2 a e calcula ed and shown in Figu e 3. Figu e 1. Fuzzy G aph G1. Ma hema ics 2023, 11, x FOR PEER REVIEW 4 o 20 3. Ope a ions on Fuzzy G aphs Using Edge Colo ing Mahaba h a e al. [29] in oduced De ini ion 11 o de e mine he weigh o colo s in a uzzy g aph. By ocusing on he op imali y o he uzzy ne wo k, we can compa e he o mula gi en in De ini ion 11 wi h he sum o he minimal membe ship alue o each colo used in he uzzy ne wo k. Hence, his esea ch de ines and modi ies he o mula in e ms o he sum o he minimum alue o he edge membe ship alues o each colo ac- co ding o he uzzy ne wo k, which yields he op imum alue o he c ea ed ne wo k and helps us de e mine how effec i e i is. The ch oma ic numbe o a uzzy g aph ep esen s he minimum numbe o colo s equi ed o colo he g aph. Le us assume ha M min is he minimum numbe o colo s used o colo he g aph. The deg ee o membe ship o such c isp g aphs is no sufficien o de e mine he s eng h o edges, and hence, some weigh is associa ed wi h he ch oma ic numbe . These weigh s o he edges can in luence he colo ing p ocess by indica ing he s eng h o associa ion o an edge wi h a pa icula colo . The weigh ed minimum o basic colo s used is deno ed as W min and is de ined as {}  = =M pp c q e gW 1 )(min min 3.1. Residue P oduc o Two Fuzzy G aphs Le 𝑅𝐹=(𝜎   ,𝜇  ) and 𝑅𝐹=(𝜎   ,𝜇  ) be wo uzzy g aph ne wo ks o c isp g aphs 𝐺  =𝑉   ,𝐸   and 𝐺  =𝑉   ,𝐸  , espec i ely. Then, i s esidue p oduc 𝑅𝐹∙𝑅𝐹 =(𝜎 ∙𝜎 ,𝜇∙𝜇) is de ined as (i) ∀ (a, b) ∈ V 1 × V 2 , (𝜎∙ 𝜎)(𝑎,𝑏)= 𝜎(𝑎) ∧ 𝜎(𝑏). (ii) ∀ (a, b) ∈ E 1 and c  w ∈ V 2 , (𝜇∙ 𝜇)((𝑎,𝑐),(𝑏,𝑤))= 𝜇 (𝑎,𝑏). 3.1.1. Example G 1 = RF 1 and G 2 = RF 2 a e he wo uzzy g aphs o 𝐺  =𝑉   ,𝐸   and 𝐺  = 𝑉  ,𝐸  , depic ed in Figu e 1 and Figu e 2, espec i ely. Then, he esidue p oduc o he uzzy ne wo k is deno ed by RF 1 ·RF 2 , as shown in Figu e 3. Figu e 1. Fuzzy G aph G 1 . Figu e 2. Fuzzy G aph G 2 . Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed RF 1 ·RF 2 a e calcula ed and shown in Figu e 3. Figu e 2. Fuzzy G aph G2. Ma hema ics 2023, 11, x FOR PEER REVIEW 5 o 20 Figu e 3. Edge membe ship alue o RF1·RF2. F om Figu e 3, he minimum numbe o basic colo s used in RF1·RF2 is 4. The weigh minimum numbe o basic colo s used in cons uc ed esidue p oduc RF1·RF2 = 0.16 + 0.2 + 0.16 + 0.16 = 0.68. Thus, he Wmin o RF1·RF2 is 0.68. 3.1.2. Find he Weigh o he Minimal Spanning T ee Using K uskal’s Algo i hm To ind he minimum spanning ee (MST) using he gi en se o edges, we ollow he ollowing s eps: Table 1 shows he weigh o he g aph, and Table 2 so s he edges in ascending o de based on hei weigh . We begin by adding he edge ay-bz wi h a speci ic weigh o he MST. Nex , we add he edge by-az o he MST wi h a weigh o 0.16. This edge does no c ea e a cycle wi hin he MST. Mo ing on, we include he edge by-cz wi h a weigh o 0.16 in he MST. This edge also does no c ea e any cycles. We con inue by adding he edge cy- bz, wi h a weigh o 0.16, o he MST. Once again, his edge main ains he p ope y o no c ea ing a cycle. Ano he edge, cy-dz, wi h a weigh o 0.2, is added o he MST wi hou causing any cycles. The edge dy-cz, wi h a weigh o 0.2, is added o he MST, ensu ing ha no cycles a e o med. We p oceed by adding he edge ax-bzy, weigh 0.28, o he MST. The inclusion o his edge does no esul in any cycles. The edge cx-by, ha ing a weigh o 0.28, is included in he MST wi hou c ea ing cycles. Mo ing o wa d, we add he edge bx-cy wi h a weigh o 0.33 o he MST; no cycles a e in oduced by his inclusion. Simi- la ly, he edge dx-cy wi h weigh 0.33 is in eg a ed in o he MST wi hou c ea ing any cycles. Las ly, we come ac oss he edge ax-bz wi h weigh 0.42. Howe e , adding his edge would c ea e a cycle wi hin he MST. Thus, we disca d i . Ha ing isi ed all he nodes and ensu ing ha he numbe o edges is ewe han he numbe o nodes, we can conclude ha he algo i hm can now be s opped. Table 1. The weigh o a gi en g aph (Figu e 4). Figu e 3. Edge membe ship alue o RF1·RF2. Ma hema ics 2023,11, 4019 5 o 16 Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed RF 1· RF 2 a e calcula ed and shown in Figu e 3. F om Figu e 3, he minimum numbe o basic colo s used in RF 1· RF 2 is 4. The weigh minimum numbe o basic colo s used in cons uc ed esidue p oduc RF 1· RF 2 = 0.16 + 0.2 + 0.16 + 0.16 = 0.68. Thus, he Wmin o RF1·RF2is 0.68. 3.1.2. Find he Weigh o he Minimal Spanning T ee Using K uskal’s Algo i hm To ind he minimum spanning ee (MST) using he gi en se o edges, we ollow he ollowing s eps: Table 2shows he weigh o he g aph, and Table 1so s he edges in ascending o de based on hei weigh . We begin by adding he edge ay-bz wi h a speci ic weigh o he MST. Nex , we add he edge by-az o he MST wi h a weigh o 0.16. This edge does no c ea e a cycle wi hin he MST. Mo ing on, we include he edge by-cz wi h a weigh o 0.16 in he MST. This edge also does no c ea e any cycles. We con inue by adding he edge cy-bz, wi h a weigh o 0.16, o he MST. Once again, his edge main ains he p ope y o no c ea ing a cycle. Ano he edge, cy-dz, wi h a weigh o 0.2, is added o he MST wi hou causing any cycles. The edge dy-cz, wi h a weigh o 0.2, is added o he MST, ensu ing ha no cycles a e o med. We p oceed by adding he edge ax-bzy, weigh 0.28, o he MST. The inclusion o his edge does no esul in any cycles. The edge cx-by, ha ing a weigh o 0.28, is included in he MST wi hou c ea ing cycles. Mo ing o wa d, we add he edge bx-cy wi h a weigh o 0.33 o he MST; no cycles a e in oduced by his inclusion. Simila ly, he edge dx-cy wi h weigh 0.33 is in eg a ed in o he MST wi hou c ea ing any cycles. Las ly, we come ac oss he edge ax-bz wi h weigh 0.42. Howe e , adding his edge would c ea e a cycle wi hin he MST. Thus, we disca d i . Ha ing isi ed all he nodes and ensu ing ha he numbe o edges is ewe han he numbe o nodes, we can conclude ha he algo i hm can now be s opped. Table 1. The edges, so ed by weigh in ascending o de . Edge ay-bz by-az by-cz cy-bz cy-dz dy-cz ax-by cx-by bx-cy dx-cy ax-bz Weigh 0.16 0.16 0.16 0.16 0.2 0.2 0.28 0.28 0.33 0.33 0.42 Ma hema ics 2023, 11, x FOR PEER REVIEW 6 o 20 Edge ax-by ax-bz bx-cy cx-by dx-cy ay-bz by-az by-cz cy-bz cy-dz dy-cz Weigh 0.28 0.42 0.33 0.28 0.33 0.16 0.16 0.16 0.16 0.2 0.2 Figu e 4. Minimum Spanning T ee o RF1·RF2. Table 2. The edges, so ed by weigh in ascending o de . Edge ay-bz by-az by-cz cy-bz cy-dz dy-cz ax-by cx-by bx-cy dx-cy ax-bz Weigh 0.16 0.16 0.16 0.16 0.2 0.2 0.28 0.28 0.33 0.33 0.42 Using K uskal’s algo i hm, he weigh o he minimal spanning ee o RF1··RF2 is shown o be 2.68. 3.1.3. Lowe Domina ion Numbe o RF1.RF2 I is possible o he e o be mo e han one minimal domina ing se in an es ablished ne wo k. The se wi h he lowes domina ion numbe among all minimal domina ing se s is conside ed o be he c ea ed ne wo k’s lowes domina ion numbe , which allows us o es he ne wo k’s op imali y. Le 𝐺 =(𝜎 ,𝜇) be he uzzy g aph o he c isp g aph GDN = (VDN, EDN), and le DN DNDN SSS ,...,, 21 be he minimal domina ing se o GDN. Fi- nally, le he co esponding domina ing numbe be deno ed by DNDNDNDN γ γ γ γ ,..., 3 , 2 , 1. Among his, he lowes ca dinali y o he domina ion num- be is called lowe domina ion numbe and is deno ed by . LDN γ Le 21212121 .)4(.)3(.)2(.)1( ,, RFRFRFRFRFRFRFRF SandSSS be he sum o he minimal dom- ina ing se o RF1.RF2 ( e e o Figu e 3) },,,{ 2 . 1 )1( dycybyayS RFRF =; },,,{ 2 . 1 )2( dxcxbxaxS RFRF =; Figu e 4. Minimum Spanning T ee o RF1·RF2. Ma hema ics 2023,11, 4019 6 o 16 Table 2. The weigh o a gi en g aph (Figu e 4). Edge ax-by ax-bz bx-cy cx-by dx-cy ay-bz by-az by-cz cy-bz cy-dz dy-cz Weigh 0.28 0.42 0.33 0.28 0.33 0.16 0.16 0.16 0.16 0.2 0.2 Using K uskal’s algo i hm, he weigh o he minimal spanning ee o RF 1·· RF 2 is shown o be 2.68. 3.1.3. Lowe Domina ion Numbe o RF1.RF2 I is possible o he e o be mo e han one minimal domina ing se in an es ablished ne wo k. The se wi h he lowes domina ion numbe among all minimal domina ing se s is conside ed o be he c ea ed ne wo k’s lowes domina ion numbe , which allows us o es he ne wo k’s op imali y. Le GDN =(σDN,µDN ) be he uzzy g aph o he c isp g aph G DN = (V DN ,E DN ), and le SDN1 , SDN2 , . . . , SDN be he minimal domina ing se o G DN . Fi- nally, le he co esponding domina ing numbe be deno ed by γDN1 , γDN2 , γDN3 , . . . , γDN . Among his, he lowes ca dinali y o he domina ion numbe is called lowe domina ion numbe and is deno ed by γLDN. Le S(1)RF1.RF2 , S(2)RF1.RF2 , S(3)RF1.RF2and S(4)RF1.RF2 be he sum o he minimal dom- ina ing se o RF1.RF2( e e o Figu e 3) S(1)RF1.RF2={ay,by,cy,dy};S(2)RF1.RF2={ax,bx,cx,dx}; S(3)RF1.RF2={az,bz,cz,dz} γ(1)RF1.RF2o S(1)RF1.RF2=0.3 +0.6 +0.6 +0.3 =1.8 γ(2)RF1.RF2o S(2)RF1.RF2=2.6. γ(2)RF1.RF2o S(3)RF1.RF2=2.1. γLDN−RF1.RF2o RF1RF2is 1.8. 3.2. Symme ic Di e ence o Two Fuzzy G aphs Le SDF 1 = (σec1,µec1) and SDF 2 = (σec2,µec2) be wo uzzy g aphs o c isp g aphs GSDF1=(Vec1,Eec1)andGSDF2=(Vec2,Eec2) , espec i ely. Then, he symme ic di e ence be ween SDF 1 and SDF 2 is deno ed by SDF 1⊕ SDF 2 = ( σec1⊕σec2 , µec1⊕µec2 ) and is de ined as ollows 1. ∀(a,b)∈V1×V2, σSDF1⊕σSDF2(a,b)=σSDF1(a)∧σSDF2(b). 2. ∀a∈V1and (b,c)∈E2(µSDF1⊕µSDF2(a,b),(a,c)=σSDF1(a)∧µSDF2(b,c). 3. ∀a∈V2and (b,c)∈E1(µSDF1⊕µSDF2((b,a),(c,a)) =µSDF1(b,c)∧σSDF2(a). 4. ∀(a,b)/∈E1and (c,w)∈E2 , (µSDF1⊕µSDF2((a,c),(b,w)) =min{σSDF1(a) , σSDF1(b),µSDF2(c,w)}. 5. ∀(a,b)∈E1and (c,w)/∈E2 , (µSDF1⊕µSDF2((a,c),(b,w)) =min{µSDF1(a,b) , σSDF2(c),σSDF2(w)}. 3.2.1. Example Le G 1 =SDF 1 and G 2 =SDF 2 be he wo uzzy g aphs o c isp g aphs GSDF1=(Vec1,Eec1)and GSDF2=(Vec2,Eec2), depic ed in Figu es 1and 2, espec i ely. The symme ic di e ence o uzzy ne wo k SDF1⊕SDF2is shown in Figu e 5. Ma hema ics 2023,11, 4019 7 o 16 Ma hema ics 2023, 11, x FOR PEER REVIEW 8 o 20 Figu e 5. Edge Membe ship alue o SDF1 ⊕ SDF2. As shown in Figu e 5, he minimum numbe o basic colo s used in SDF1 ⊕ SDF2 is 7. The weigh minimum numbe o basic colo s used in he cons uc ed symme ic diffe - ence ne wo k SDF1 ⊕ SDF2 = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.75 + 0.5 = 3.75. Thus, he Wmin o SDF1 ⊕ SDF2 is 3.75. Using K uskal’s algo i hm, he weigh o he minimal spanning ee o SDF1 ⊕ SDF2 was ound o be 7. 3.2.2. Lowe Domina ion Numbe o SDF1 ⊕ SDF2 Le 21 )4( 21 )3( , 21 )2( , 21 )1( SDFSDF Sand SDFSDF S SDFSDF S SDFSDF S⊕⊕⊕⊕ be some o he minimal domina ing se o SDF1 ⊕ SDF2 (see Figu e 5) },{ 21 )1( ayax SDFSDF S= ⊕; },{ 21 )2( ayaz SDFSDF S= ⊕; },{ 21 )3( dydx SDFSDF S= ⊕; }.,{ 21 )4( dzdy SDFSDF S= ⊕ 2 . 1 )1( SDFSDF ⊕ γ o = ⊕21 )1( SDFSDF S 2.2 2 . 1 )2( SDFSDF ⊕ γ o .6.1 21 )2( = ⊕SDFSDF S 2 . 1 )3( SDFSDF ⊕ γ o .2.2 21 )3( = ⊕SDFSDF S 2 . 1 )4( SDFSDF ⊕ γ o .9.1 21 )4( = ⊕SDFSDF S Figu e 5. Edge Membe ship alue o SDF1⊕SDF2. Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed SDF1⊕SDF2a e calcula ed, as shown in Figu e 5. As shown in Figu e 5, he minimum numbe o basic colo s used in SDF 1⊕ SDF 2 is 7. The weigh minimum numbe o basic colo s used in he cons uc ed symme ic di e ence ne wo k SDF 1⊕ SDF 2 = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.75 + 0.5 = 3.75. Thus, he W min o SDF1⊕SDF2is 3.75. Using K uskal’s algo i hm, he weigh o he minimal spanning ee o SDF 1⊕ SDF 2 was ound o be 7. 3.2.2. Lowe Domina ion Numbe o SDF1⊕SDF2 Le S(1)SDF1⊕SDF2 , S(2)SDF1⊕SDF2 , S(3)SDF1⊕SDF2and S(4)SDF1⊕SDF2 be some o he mini- mal domina ing se o SDF1⊕SDF2(see Figu e 5) S(1)SDF1⊕SDF2={ax,ay};S(2)SDF1⊕SDF2={az,ay}; S(3)SDF1⊕SDF2={dx,dy};S(4)SDF1⊕SDF2={dy,dz}. γ(1)SDF1.⊕SDF2o S(1)SDF1⊕SDF2=2.2 γ(2)SDF1.⊕SDF2o S(2)SDF1⊕SDF2=1.6. γ(3)SDF1.⊕SDF2o S(3)SDF1⊕SDF2=2.2. γ(4)SDF1.⊕SDF2o S(4)SDF1⊕SDF2=1.9. γLDN−SDF1⊕.SDF2o SDF1⊕SDF2is 1.6. Ma hema ics 2023,11, 4019 8 o 16 3.3. Max P oduc o Two Fuzzy G aphs Le MF 1 = (σm 1,µm 1and MF2=σm 2,µm 2 be wo uzzy ne wo ks o c isp g aphs GMF1=Vm 1,Em 1and GMF2=Vm 2,Em 2 , espec i ely. The maximal p od- uc o uzzy g aphs MF 1 and MF 2 is ep esen ed by MF 1 * MF 2 = (σm 1 * σm 2 , µm 1 *, µm 2 ) and is de ined as: (i) ∀(a,b)∈Vm 1×Vm 2,σm 1*σm 2(a,b)=σm 1(a)∨σm 2(b). (ii) ∀a∈Vm 1and (b,c)∈Em 2,(µm 1*µm 2((a,b),(a,c)) =σm 1(a)∨µm 2(b,c). (iii) ∀a∈Vm 2and (b,c)∈Em 1,µm 1*µm 2((b,a),(c,a)) =µm 1(b,c)∨σm 2(a). 3.3.1. Example G 1 =MF 1 and G 2 =MF 2 de ines he wo uzzy g aphs o c isp g aphs GMF1=(Vec1,Eec1)and GMF2=(Vec2,Eec2) , depic ed in Figu es 1and 2, espec i ely. The max p oduc o he uzzy ne wo k MF1* MF2is shown in Figu e 6. Ma hema ics 2023, 11, x FOR PEER REVIEW 9 o 20 2 . 1SDFSDFLDN ⊕− γ o SDF1 ⊕ SDF2 is 1.6. 3.3. Max P oduc o Two Fuzzy G aphs Le MF1 = (𝜎,𝜇) and 𝑀𝐹=(𝜎 ,𝜇) be wo uzzy ne wo ks o c isp g aphs 𝐺=𝑉 ,𝐸 and 𝐺=𝑉 ,𝐸, espec i ely. The maximal p oduc o uzzy g aphs MF1 and MF2 is ep esen ed by MF1 * MF2 = (𝜎 * 𝜎, 𝜇*,𝜇) and is de- ined as: (i) ∀ (a, b) ∈ 𝑉 × 𝑉, 𝜎∗ 𝜎(𝑎,𝑏)= 𝜎(𝑎) ∨ 𝜎(𝑏). (ii) ∀ a ∈𝑉 and (b, c) ∈𝐸, (𝜇∗ 𝜇)((𝑎,𝑏),(𝑎,𝑐))= 𝜎(𝑎) ∨ 𝜇(𝑏,𝑐). (iii) ∀ a ∈𝑉 and (b, c) ∈𝐸, 𝜇∗ 𝜇(𝑏,𝑎),(𝑐,𝑎)= 𝜇(𝑏,𝑐)∨ 𝜎(𝑎). 3.3.1. Example G1 = MF1 and G2 = MF2 de ines he wo uzzy g aphs o c isp g aphs 𝐺= 𝑉,𝐸 and 𝐺=𝑉 ,𝐸, depic ed in Figu e 1 and Figu e 2, espec i ely. The max p oduc o he uzzy ne wo k MF1 * MF2 is shown in Figu e 6. Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed MF1 * MF2 we e calcula ed, as shown in Figu e 6. Figu e 6. Edge Membe ship alue o MF1 * MF2. As shown in Figu e 6, he minimum numbe o basic colo s used in MF1 * MF2 is 4. The weigh minimum numbe o basic colo s used in he cons uc ed maximal p oduc ne wo k MF1 * MF2 = 0.37 + 1 + 0.83 + 0.83 = 3.03. Thus, he Wmin o MF1 * MF2 is 3.03. Using K uskal’s algo i hm, he weigh o he minimal spanning ee o MF1 * MF2 was ound o be 9.29. Figu e 6. Edge Membe ship alue o MF1*MF2. Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed MF 1 *MF 2 we e calcula ed, as shown in Figu e 6. As shown in Figu e 6, he minimum numbe o basic colo s used in MF 1 * MF 2 is 4. The weigh minimum numbe o basic colo s used in he cons uc ed maximal p oduc ne wo k MF1* MF2= 0.37 + 1 + 0.83 + 0.83 = 3.03. Thus, he Wmin o MF1* MF2is 3.03. Using K uskal’s algo i hm, he weigh o he minimal spanning ee o MF 1 * MF 2 was ound o be 9.29. 3.3.2. Lowe Domina ion Numbe o MF1*MF2 Le S(1)MF1∗MF2 , S(2)MF1∗MF2 , S(3)MF1∗MF2and S(4)MF1∗MF2 be some o he minimal dom- ina ing se o MF1* MF2(see Figu e 6) S(1)MF1∗MF2={ay,cx,cz,dy}S(2)MF1∗MF2={ay,bx,by,dy}; Ma hema ics 2023,11, 4019 9 o 16 S(3)MF1∗MF2={bx,cx,ay,dy};S(4)MF1∗MF2={ax,bz,cx,dz}. γ(1)MF1.∗MF2o S(1)MF1∗MF2=6.5;γ(2)MF1.∗MF2o S(2)MF1∗MF2=6.7; γ(3)SMF1.∗MF2o S(3)MF1∗MF2=7.2;γ(4)MF1.∗MF2o S(4)MF1∗MF2=5.7; γLDN−MF1.∗MF2o MF1∗MF2is 5.7. 3.4. Lexicog aphic P oduc o Two Fuzzy G aphs Le LF 1 = (M 1 , P 1 ) and LF 2 = (M 2 , P 2 ) be wo uzzy g aphs o he c isp g aphs GLF1=(Vec1,Eec1)and GLF2=(Vec2,Eec2) espec i ely. The lexicog aphic p oduc o he wo g aphs is deno ed as LF 1· LF 2 in uzzy g aph pai (M,P), such ha (i) M(a1,b2)=min(M1(a1),M2(b2)),∀(a1,b2)∈M1×M2. (ii) P((x,b2)(x,d2)=min(M1(x),M2(b2d2),∀x∈M1,b2d2∈P2. (iii) P((a1,b2)(c1,d2)) =min(P1(a1c1),P2(b2d2)),∀a1b1∈P1and b2d2∈P2. 3.4.1. Example Le G 1 =LF 1 and G 2 =LF 2 be wo uzzy g aphs o c isp g aphs GLF1=(Vec1,Eec1) and GLF2=(Vec2,Eec2) , depic ed in Figu es 1and 2, espec i ely. The lexicog aphic p oduc o he uzzy ne wo k is deno ed by LF1·LF2, as shown in Figu e 7. Ma hema ics 2023, 11, x FOR PEER REVIEW 11 o 20 Figu e 7. Edge Membe ship alue o LF1·LF2. As shown in Figu e 7, he minimum numbe o basic colo s used in LF1·LF2 is 6. The weigh minimum numbe o basic colo s used in he cons uc ed lexicog aphic p oduc o ne wo k LF1·LF2 = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3. Thus, he Wmin o LF1·LF2 is 3. Using K uskal’s algo i hm, he weigh o he minimal spanning ee o LF1·LF2 was ound o be 5.5. 3.4.2. Lowe Domina ion Numbe o LF1·LF2 Le 2 * 1LFLF S be he minimal domina ing se (only one domina ing se ) o LF1·LF2 (see Figu e 7) }.,,,{ 2 * 1 dycybyay LFLF S=2 *. 1LFLFLDN − γ o .3 2 * 1is LFLF S Figu e 7. Edge Membe ship alue o LF1·LF2. 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